Basic Topological Group Theory
This first chapter is an introduction to the basic structural facts in the theory of
topological groups. There are no surprises. The basic analytic facts are in Chapter
3. The concepts here are essential for the rest of this monograph.
1.1. Definition and Separation Properties
A topological group is a group with a topology such that the group opera-
tions are continuous. In other words, the algebraic and topological structures are
mutually consistent. We make the formal definition.
1.1.1. Let G be both an abstract group and a topological space.
Then G is a topological group if
the one point subsets of G are closed
the map G x G G by (g, ft) i—
is continuous. (}
1.1.2. Now let G be a topological group. Notice that the maps G » G by
g h-» g~x and G x G » G by (g, ft) i— gh are continuous. For the first, (ft, g) H-» hg~l
is continuous and we set ft = 1. For the second, Gx G * G by (g, /i) i— #/i now is the
composition (g, ft) i-» (g,
i-» g(/i
- 1
- 1
of two continuous maps. Similarly these
two conditions imply continuity of (g, ft) i—
So together they are equivalent
to the second continuity condition of the definition of topological group.
1.1.3. If {#i,... ,gn} C G and {ri,... , rn} are integers, and if V is a neigh-
borhood of ft = g^1 ... g7^1 in G, then there are neighborhoods Uj of gj such that
t/!""1 . . . ^ c y . For the map Gx--xG-G, given by (gu ..., #n) H- ^ ... # ^ ,
is continuous. This fact is crucial to many arguments in topological group theory.
If G = (Rn, +) it corresponds to "e/n" arguments from elementary calculus.
1.1.4. The left translations £g : h *-» gh on G, the right translations
rg : h y-+ hg, and the conjugations ag : ft, H-
all are homeomorphisms
of G. In effect, they are continuous because joint continuity of (-U, t) i— uv, of
(ix,v) i— vu, and of (u,v) i—
each implies separate continuity, and because
=rg-i, and a"
= a^-i.
LEMMA 1.1.5. A topological group is a regular2 topological space.
This is the Tychonoff separation condition T\.
A topological space X is regular if, given a closed subset F C X and a point x £ F, there
are disjoint open subsets U, V C X with x £ U and F C V. This is the Tychonoff separation
condition T3.
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